Least Action Principle of Crystal Formation of Dense Packing Type & the Proof of Kepler's Conjecture

The dense packing of microscopic spheres (i.e. atoms) is the basic geometric arrangement in crystals of mono-atomic elements with weak covalent bonds, which achieves the optimal "known density" of B/√18. In 1611, Johannes Kepler had already "conjectured" that B/√18 should be the optimal "density" of sphere packings. Thus, the central problems in the study of sphere packings are the proof of Kepler's conjecture that B/√18 is the optimal density, and the establishing of the least action principle that the hexagonal dense packings in crystals are the geometric consequence of optimization of density. This important book provides a self-contained proof of both, using vector algebra and spherical geometry as the main techniques and in the tradition of classical geometry. Contents Foreword Acknowledgment List of Symbols Chapter 1 Introduction 1.1 Sphere Packings and the Sphere Packing Problem 1.1.1 Density of an infinite packing 1.2 Kepler’s Conjecture on Sphere Packings 1.2.1 Another mathematical formulation of the sphere packing problem and Kepler’s Conjecture 1.3 Density of Finite Packings without a Container and the Least Action Principle of Crystal Formation 1.3.1 Hexagonal close packings 1.3.2 Least action principle and crystal formation 1.3.3 Local cell decomposition, local density and the Dodecahedron Conjecture 1.4 Locally Averaged Density and the Intrinsic Density of the Second Kind 1.5 Main Theorems on Sphere Packings 1.5.1 Relationship between the two kinds of relative densities and the proof of Kepler’s Conjecture Chapter 2 The Basics of Euclidean and Spherical Geometries and a New Proof of the Problem of Thirteen Spheres 2.1 Vector Algebra and Basic Spherical Geometry 2.1.1 Basic properties of the unit sphere 2.1.2 Vector algebra and spherical trigonometry 2.1.3 Some further results on areas of spherical triangles and quadrilaterals 2.1.3.1 Spherical quadrilaterals 2.1.3.2 Shearing deformations 2.2 Spherical Configurations, Area Estimates and a New Proof of the Impossibility of Thirteen Touching Neighbors 2.2.1 Examples of problems on the distribution of point on S2(1) 2.2.2 Spherical configuration and some basic techniques of area estimation 2.2.3 Techniques of area estimates 2.2.4 Star configurations 2.2.5 Another proof of the impossibility of thirteen touching neighbors Chapter 3 Circle Packings and Sphere Packings 3.1 The Problem of Circle Packings 3.2 Sphere Packings and Crystal Formations, Kepler’s Conjecture and a Least Action Principle of Crystal Formation 3.2.1 Three kinds of sphere packings and the concepts of their densities 3.2.1.1 Some natural problems on the density of the first kind and a generalized Dodecahedron Conjecture 3.2.2 The sphere packing problem and Kepler’s Conjecture 3.2.2.1 A brief comparison between the sphere packing and the circle packing problem 3.2.3 Sphere packings and crystal formation 3.2.3.1 The least action principle and crystal formation 3.2.4 The least action principle of crystal formation and the localization of the proof of Kepler’s Conjecture 3.3 Some Basic Ideas and Crucial Understanding on which the Proofs of the Major Theorems are Based 3.3.1 Volume estimates of local cells 3.3.1.1 Basic geometry of local cells 3.3.1.2 Basic strategy of volume estimations of local cells 3.3.1.3 Volume estimation techniques specifically developed for local cells 3.3.1.4 Some examples of volume estimation 3.3.2 Geometry of single-layer local packings 3.3.2.1 Quantitative refinements of the problem of thirteen spheres 3.3.2.2 The non-tightness of local packings with twelve touching or almost touching neighbors 3.3.3 Geometry of double-layer local packing Chapter 4 Geometry of Local Cells and Specific Volume Estimation Techniques for Local Cells 4.1 Basic Geometry of Local Packings and Local Cells 4.1.1 Associated spherical configurations and polyhedrons of (E, {hj}) 4.1.2 Geometric correlations between spherical configurations and associated polyhedrons 4.1.2.1 The correlation between S and T 4.1.2.2 The basic geometry of T 4.1.2.3 The relationships between T and T (resp. S* and S* 4.1.2.4 The peripheral part and its rectilinear slabs 4.1.3 Basic strategies of volume estimation of local cells 4.1.3.1 The separation of a local cell into its core part and its peripheral part 4.1.3.2 The subdivision of the peripheral part into rectilinear slabs 4.1.3.3 Spherical configuration and a basic volume formula of T, the basic strategy for the estimation of the core part 4.2 Technique of Volume Estimation of the Core Part 4.2.1 The volume function of tangent subpolyhedron 4.2.1.1 A basic volume formula 4.2.2 A volume formula of T 4.2.2.1 A gradient formula of vol T 4.2.3 Two basic lemmas of volume estimation 4.2.4 Some direct applications 4.3 Volume Estimation of a Rectilinear Slab 4.3.1 The local geometry surrounding a given neighbor of L‘(S0) 4.3.2 Lower bound volume estimates of rectilinear slabs 4.4 Volume Estimation of Local Cells Chapter 5 Estimates of Total Buckling Height 5.1 The Correlation Between Buckling Heights and Area Estimates 5.2 Area Estimates of Star Configurations 5.2.1 Area decreasing deformations of star configurations 5.2.2 Area estimates of star configurations 5.2.2.1 Examples of non-deformable 6-stars with a given set of edge-length bounds 5.2.2.2 7-star with individual lower bounds on edge-lengths close to 3 5.3 Estimation of Total Buckling Height 5.3.1 The proof of Lemma 5.3.3 5.3.2 The proof of Lemma 5.3.2 5.3.3 The proof of Lemma 5.3.1 Chapter 6 The Proof of the Dodecahedron Conjecture 6.1 The Proof of Case 1: m = 12 6.2 The Proof of Case 2: m < 11 6.3 The Proof of Case 3: 13 < m 4 References Index The dense packing of microscopic spheres (i.e. atoms) is the basic geometric arrangement in crystals of mono-atomic elements with weak covalent bonds, which achieves the optimal "known density" of p/v18. In 1611, Johannes Kepler had already "conjectured" that p/v18 should be the optimal "density" of sphere packings. Thus, the central problems in the study of sphere packings are the proof of Kepler's conjecture that p/v18 is the optimal density, and the establishing of the least action principle that the hexagonal dense packings in crystals are the geometric consequence of optimization of densi This work provides proof of Kepler's conjecture that B/O18 is the optimal density, and establishes the least action principle, which states that the hexagonal dense packings in crystals are the geometric consequence of optimization of density. Wu-yi Hsiang. Includes Bibliographical References (p. 397-399) And Index.